For further trailing digits like hundredths place digit and so on we take remainders with next powers of 10 like 1000 for hundredths place digit with probably 125 and 8 used for Chinese Remainder Theorem

Patterns and Generalizations for Cyclicity

For odd numbers

(Any odd number with unit digit 1, 3, 7, 9)^20N will have 01 as the last two digitsexample Last two digits of 37^20 = Last two digits of 37^40 = Last two digits of 37^60

(Any odd number with unit digit 5)^N where N >1 will have 25 as the last two digits